$L^ p$ function decomposition on $C^ \infty$ totally real submanifolds of $\textbf {C}^ n$

Abstract:

For $1 < p < \infty$ we show that ${L^p}$ functions defined on a ${C^\infty }$ totally real submanifold of ${\mathbb {C}^n}$ can be locally decomposed into the sum of boundary values of holomorphic functions in wedges such that the boundary values are in ${L^p}$. The general case of a ${C^\infty }$ totally real submanifold is reduced to the flat case of ${\mathbb {R}^n}$ in ${\mathbb {C}^n}$ by an almost analytic change of variables. ${L^p}$ results in the flat case are then obtained using Fourier multipliers. In transporting these results back to the manifold we lose analyticity, so it is necessary to solve a $\overline \partial$ problem in an appropriate domain. This gives holomorphy in the wedges but produces a ${C^\infty }$ error on the edge. This ${C^\infty }$ function is then holomorphically decomposed using the FBI transform with a careful analysis to check that the functions are ${C^\infty }$ up to the edge and do not destroy the ${L^p}$ behavior.